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To convert a fixed-point number to floating-point, one may convert the integer to floating-point and then divide it by the scaling factor S. This conversion may entail rounding if the integer's absolute value is greater than 2 24 (for binary single-precision IEEE floating point) or of 2 53 (for double-precision).
The Q notation is a way to specify the parameters of a binary fixed point number format. For example, in Q notation, the number format denoted by Q8.8 means that the fixed point numbers in this format have 8 bits for the integer part and 8 bits for the fraction part. A number of other notations have been used for the same purpose.
Similar binary floating-point formats can be defined for computers. There is a number of such schemes, the most popular has been defined by Institute of Electrical and Electronics Engineers (IEEE). The IEEE 754-2008 standard specification defines a 64 bit floating-point format with: an 11-bit binary exponent, using "excess-1023" format.
Two's complement is the most common method of representing signed (positive, negative, and zero) integers on computers, [1] and more generally, fixed point binary values. Two's complement uses the binary digit with the greatest value as the sign to indicate whether the binary number is positive or negative; when the most significant bit is 1 the number is signed as negative and when the most ...
Fixed-point arithmetic, for an alternative approach at computation with rational numbers (especially beneficial when the exponent range is known, fixed, or bound at compile time) IBM System z9 , the first CPU to implement IEEE 754-2008 decimal arithmetic (using hardware microcode)
However, decimal fixed-point and decimal floating-point formats are still important and continue to be used in financial, commercial, and industrial computing, where the subtle conversion and fractional rounding errors that are inherent in binary floating point formats cannot be tolerated. [11]
Fixed-point representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it can be used to perform normal ...
A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23 ) × 2 127 ≈ 3.4028235 ...